Following [1]}, [2]}, [3]}, since the equality constraints (REF )(REF )(REF ) are affine, and the inequality constraint (REF ) is convex, this problem is a convex program, then we can use strong duality of convex programs, which indicates that under certain mild conditions (so-called Slater’s condition) the dual program has the same optimal value as the primal.
We will choose the boundary condition (REF ) and (REF ) to be \(implicit\)  and the divergenceless constraint (REF ) and norm bound (REF ) to be \(explicit\) . This means that we introduce Lagrange multipliers, \({\psi _{ij}}\)  (\(i < j\) ) and \(\lambda \)  (subject to \(\lambda \ge 0\) ) for constraints (REF ), (REF ) respectively. We then solve the maximization problem without imposing these two constraints, but while still imposing constraints (REF ) and (REF ). Adding the Lagrange multiplier terms to the objective, we arrive at the following Lagrangian functional:
\(L = \sum \limits _{i < j} {2\int _i {{{\vec{v}}_{ij}}} } + \int _C {\sqrt{g} \left( {\sum \limits _{i < j} {{\psi _{ij}}\nabla \cdot {{\vec{v}}_{ij}}} + \lambda \left( {\frac{1}{{4{G_N}}} - \left| {{{\vec{v}}_{ij}}} \right|} \right)} \right)}. \) 
